scipy.special.eval_sh_jacobi

scipy.special.eval_sh_jacobi(n, p, q, x, out=None) = <wrapped_ufunc 'eval_sh_jacobi'>

Evaluate shifted Jacobi polynomial at a point.

Defined by

\[G_n^{(p, q)}(x) = \frac{\Gamma(n+1)\Gamma(n+p)}{\Gamma(2n+p)} P_n^{(p - q, q - 1)}(2x - 1),\]

where \(P_n^{(\cdot, \cdot)}\) is the n-th Jacobi polynomial. See 22.5.2 in [AS] (or equivalently [DLMF]) for details. Note that here, in contrast to eval_sh_legendre, eval_sh_chebyt, and eval_sh_chebyu, more than just the argument is shifted.

Parameters:

nfloat

Degree of the polynomial. If not an integer, the result is determined via the relation to binom and eval_jacobi.

pfloat

Parameter, must satisfy \(p - q > -1\).

qfloat

Parameter, must be greater than 0.

xarray_like

Points at which to evaluate the polynomial.

outndarray, optional

Optional output array for the function values

Returns:

Gscalar or ndarray

Values of the shifted Jacobi polynomial.

See also

roots_sh_jacobi

roots and quadrature weights of shifted Jacobi polynomials

sh_jacobi

shifted Jacobi polynomial object

eval_jacobi

evaluate Jacobi polynomials

Notes

Array API Standard Support

eval_sh_jacobi has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	✅	⛔
JAX	✅	⛔
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but typically none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. eval_sh_jacobi does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

References

[AS]

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

[DLMF]

NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov/18.1.E2

Examples

Try it in your browser!

>>> import numpy as np
>>> from scipy.special import eval_sh_jacobi
>>> import matplotlib.pyplot as plt

These are the parameter values that we’ll use in this example.

>>> p = 0.5
>>> q = 0.125

Evaluate the order 3 polynomial at a few values of x.

>>> eval_sh_jacobi(3, p, q, [0, 0.25, 0.5])
array([-0.00344968,  0.01217532, -0.02333604])

Plot the polynomials of order 0 through 4.

>>> x = np.linspace(0, 1, 800)
>>> fig, ax = plt.subplots(1, 1, constrained_layout=True)
>>> for n in range(5):
...     y = eval_sh_jacobi(n, p, q, x)
...     ax.plot(x, y, label=f'{n = }')
>>> ax.grid()
>>> ax.set_title("Shifted Jacobi Polynomials")
>>> ax.set_xlabel("$x$")
>>> ax.set_ylabel("$G_n^{(p, q)}(x)$")
>>> ax.legend(loc='upper left', bbox_to_anchor=(1, 1), shadow=True)
>>> plt.show()

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